Page:American Anthropologist NS vol. 1.djvu/515

456

Coefficient of Correlation.

Indians of Northern Coast Shuswap Sioux of British Columbia, Indians. Indians.

Stature and Length of Head + 0.26

Stature and Breadth of Head + 0.09

Breadth of Face and Length of Head. + 0.30 + 0.28 + 0.27

Breadth of Face and Breadth of Head. + 0.53 + 0.62 + 0.52

Height of Face and Length of Head. + 0.24 + °- 2 7 + 0.30

Height of Face and Breadth of Head. — 0.10 +0.22 +0.22

If we desire to eliminate the influences of these causes from the apparent correlation between length and breadth of head, we must consider each of these a function not only of the other, but also of all the measurements the effect of which we desire to exclude. In doing so, we may pursue the same method that we used in the beginning (p. 450) and by a series of substitutions in (1) we find

(4) Average x 1 = q 123. . . . p x 2 + q 132. . . . p x 3 +. . . q 1 p 2. . (p-1)x p

The values q may be determined in the following manner. We will call

\5/ ^abc. . == '*al»c. . y.

By multiplying (4) successively with x 2 at x 3. . x p, we have

(6) r12 = r123. . . p + r 132. . . p r 32 +. . . + r 1 p 2. . (p-1) rp2

r13 = r123. . . p r 23 + r 132. . . p +. . . + r 1 p 2. . (p-1) rp3

r1p + r123. . . p r 3p +. . . + r 1 p 2. . (p-1)

By multiplying the last of these equations successively by rp2, rp3. . and subtracting from the first, second. . . equation, we find

r12 - r1prp2 = r123. . p(1 - r2p rp2 ) + r132. . p(r32 -r3prp2)

. + . . . . . . . r1(p-1)2. . . p(r(p-1)2 - r(p-1)prp2) ..